Asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three. II

Abstract: We investigate the asymptotic stability of standing waves for a model of Schr\"odinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $\delta$) interaction with strength $\alpha$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $\Phi_\omega$, and the linearization around them admits two imaginary eigenvalues which in principle could correspond to non decaying states, so preventing asymptotic relaxation towards an equilibrium orbit. Without using the Fermi Golden Rule we prove that, in the range $(\frac{1}{\sqrt 2},\sigma^*)$ for a certain $\sigma^* \in (\frac{1}{\sqrt{2}}, \frac{\sqrt{3} +1}{2 \sqrt{2}}]$, the dynamics near the orbit of a standing wave asymptotically relaxes towards a standing state. Contrarily to the main results in the field, the admitted nonlinearity is $L^2$-subcritical.